11–12Mathematics Extension 1 11–12 Syllabus

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Year 11

Year 12

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Inverse trigonometric functions

MAO-WM-01
develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly
ME1-12-03
solves problems involving inverse trigonometric functions

Inverse trigonometric functions

Definitions of inverse trigonometric functions

Graph $y = \sin x$ , $y = \cos x$ and $y = \tan x$ , for $- 2 π \leq x \leq 2 π$ using graphing applications and recognise that these three functions fail the horizontal line test
Examine possible domain restrictions of $y = \sin x$ , $y = \cos x$ and $y = \tan x$ to obtain the corresponding inverse functions using graphing applications
Define $\sin^{- 1} x$ or $\arcsin x$ to be the inverse function of $y = \sin x$ restricted to $- \frac{π}{2} \leq x \leq \frac{π}{2}$ , and determine the domain and range of ${y = \sin}^{- 1} x$
Define $\cos^{- 1} x$ or $\arccos x$ to be the inverse function of $y = \cos x$ restricted to $0 \leq x \leq π$ , and determine the domain and range of ${y = \cos}^{- 1} x$
Define $\tan^{- 1} x$ or $\arctan x$ to be the inverse function of $y = \tan x$ restricted to $- \frac{π}{2} < x < \frac{π}{2}$ , and determine the domain and range of ${y = \tan}^{- 1} x$
Evaluate and simply expressions, and prove results using the definitions of $\sin^{- 1} x$ , $\cos^{- 1} x$ and $\tan^{- 1} x$

Graphs of inverse trigonometric functions

Graph $y = \sin x$ for $- \frac{π}{2} \leq x \leq \frac{π}{2}$ , then use a reflection in $y = x$ to graph $y = \sin^{- 1} x$ with and without graphing applications
Graph $y = \cos x$ for $0 \leq x \leq π$ , then use a reflection in $y = x$ to graph $y = \cos^{- 1} x$ with and without graphing applications
Graph $y = \tan x$ for $- \frac{π}{2} < x < \frac{π}{2}$ , then use a reflection in $y = x$ to graph $y = \tan^{- 1} x$ with and without graphing applications
Classify ${y = \sin}^{- 1} x$ , $y = \cos^{- 1} x$ and $y = \tan^{- 1} x$ as odd, even, or neither odd nor even
Examine the properties $\sin^{- 1} (- x) = - \sin^{- 1} x$ , $\cos^{- 1} (- x) = π - \cos^{- 1} x$ , $\tan^{- 1} (- x) = - \tan^{- 1} x$ and $\cos^{- 1} x + \sin^{- 1} x = \frac{π}{2}$ using graphing applications and prove them algebraically
Use the properties $\sin^{- 1} (- x) = - \sin^{- 1} x$ , $\cos^{- 1} (- x) = π - \cos^{- 1} x$ , $\tan^{- 1} (- x) = - \tan^{- 1} x$ and $\cos^{- 1} x + \sin^{- 1} x = \frac{π}{2}$ to solve problems, simplify expressions and prove results
Graph the composite functions $y = \sin (\sin^{- 1} x)$ , $y = \cos (\cos^{- 1} x)$ and $y = \tan (\tan^{- 1} x)$ by considering their domains and ranges
Examine the graphs of the composite functions $y = \sin (\sin^{- 1} x)$ , $y = \cos (\cos^{- 1} x)$ and $y = \tan (\tan^{- 1} x)$ , establish the values of $x$ for which the formulas $\sin (\sin^{- 1} x) = x$ , $\cos (\cos^{- 1} x) = x$ and $\tan (\tan^{- 1} x) = x$ are true, and apply the formulas to solve problems, simplify expressions and prove results
Graph the composite functions ${y = \sin}^{- 1} (\sin x)$ , ${y = \cos}^{- 1} (\cos x)$ and ${y = \tan}^{- 1} (\tan x)$ by considering their domains and ranges
Examine the graphs of the composite functions ${y = \sin}^{- 1} (\sin x)$ , ${y = \cos}^{- 1} (\cos x)$ and ${y = \tan}^{- 1} (\tan x)$ , establish the values of $x$ for which the formulas $\sin^{- 1} (\sin x) = x$ , $\cos^{- 1} (\cos x) = x$ and $\tan^{- 1} (\tan x) = x$ are true, and apply the formulas to solve problems, simplify expressions and prove results
Use graphing applications to explore reflections, translations and dilations of the functions $f (x) = \sin^{- 1} x$ , $f (x) = \cos^{- 1} x$ and $f (x) = \tan^{- 1} x$ , and confirm that the principles of transformations hold for the inverse trigonometric functions
Apply the principles of transformations to inverse trigonometric functions to determine the function rule and graph the function, finding the domain and range of the transformed graph and determining any intercepts and asymptotes where appropriate

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11–12Mathematics Extension 1 11–12 Syllabus

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